Q: A biologist models the population of a bacteria culture with the function $$P(t) = 500e$$^{0.3t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

$$e^{1.5}$$ \approx 4.48$$, so P(5$$) \approx 500 \times 4.48 = 2240$$

Q: A car depreciates in value according to the function $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 3 years?

$$V(3) = 20000 \times (0.85)^3$ \approx 12262.5$$

Q: A medication is administered so that its concentration in the bloodstream decreases by 20% every hour. If the initial concentration is 50 mg/L, what will the concentration be after 3 hours?

$$50 \times (0.8)^3$ = 25.6 mg/L$$

Question 1

Which of the following best describes the domain of the function $$f(x) = \dfrac{1}{x^2$ - 4}$$?

Accepted Answer

$$x 
eq 2$$ and $$x 
eq -2$$

Question 2

A biologist models the population of a bacteria culture with the function $$P(t) = 500e$$^{0.3t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

Accepted Answer

$$e^{1.5}$$ \approx 4.48$$, so P(5$$) \approx 500 	imes 4.48 = 2240$$

Question 3

Which of the following is the inverse of the function $$f(x) = 2x + 7$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x - 7}{2}$$

Question 4

Solve the system of equations using any method:

$$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$

Accepted Answer

$$x = 2, y = 1$$

Question 5

Which of the following is the correct factorization of the polynomial $$x^3 - 4x$?

Accepted Answer

$$x(x^2 - 4) = x(x - 2)(x + 2$$)$$

Question 6

A radioactive substance decays according to $$A(t) = 1000e$$^{-0.05t}$$, where t$ is in years. How long will it take for the substance to decay to 500 grams?

Accepted Answer

$$t = \dfrac{\ln(0.5)}{-0.05} \approx 13.86$$ years

Question 7

Which of the following is the solution set for the inequality $$|2x - 3| \u003c 5$$?

Accepted Answer

$$-1 \u003c x \u003c 4$$

Question 8

Given $$f(x) = x^2$ and g(x) = x + 1$, what is (f$$ \circ g)(2)$$?

Accepted Answer

$$9$$

Question 9

Which of the following is the correct solution to the equation $$\log_2$(x) = 5$$?

Accepted Answer

$$x = 32$$

Question 10

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18$$ thousand dollars at $$x = 2$$ thousand units

Question 11

Which of the following is the solution to the equation $$2^{x+1} = 16$?

Accepted Answer

$$x = 3$$

Question 12

If $$f(x) = \dfrac{3x - 1}{x + 2}$$, what is the horizontal asymptote of $$f(x)$$?

Accepted Answer

$$y = 3$$

Question 13

Which of the following is the solution to the quadratic equation $$x^2 - 6x + 5 = 0$?

Accepted Answer

$$x = 1$$ and $$x = 5$$

Question 14

A car depreciates in value according to the function $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 3 years?

Accepted Answer

$$V(3) = 20000 	imes (0.85)^3$ \approx 12262.5$$

Question 15

Which of the following is the solution to the equation $$\ln(x) = 2$$?

Accepted Answer

$$x = e^2$

Question 16

Given the system $$\begin{cases} x + y = 4 \ x - y = 2 \end{cases}$$, what is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 17

Which of the following is the correct solution to the equation $$|x + 2| = 7$$?

Accepted Answer

$$x = 5 or$$ $$x = -9$$

Question 18

If $$f(x) = x^2$ - 4x + 3$$, what is the vertex of the parabola represented by $$f(x)$$?

Accepted Answer

$$(2, -1)$$

Question 19

Which of the following is the correct solution to the equation $$\log_{10}(x) = -2$$?

Accepted Answer

$$x = 0.01$$

Question 20

A medication is administered so that its concentration in the bloodstream decreases by 20% every hour. If the initial concentration is 50 mg/L, what will the concentration be after 3 hours?

Accepted Answer

$$50 	imes (0.8)^3$ = 25.6 mg/L$$

College Algebra: Functions, Equations, and Applications – Study Guide

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